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Theorem quad2 22352
Description: The quadratic equation, without specifying the particular branch  D to the square root. (Contributed by Mario Carneiro, 23-Apr-2015.)
Hypotheses
Ref Expression
quad.a  |-  ( ph  ->  A  e.  CC )
quad.z  |-  ( ph  ->  A  =/=  0 )
quad.b  |-  ( ph  ->  B  e.  CC )
quad.c  |-  ( ph  ->  C  e.  CC )
quad.x  |-  ( ph  ->  X  e.  CC )
quad2.d  |-  ( ph  ->  D  e.  CC )
quad2.2  |-  ( ph  ->  ( D ^ 2 )  =  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) )
Assertion
Ref Expression
quad2  |-  ( ph  ->  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  =  0  <-> 
( X  =  ( ( -u B  +  D )  /  (
2  x.  A ) )  \/  X  =  ( ( -u B  -  D )  /  (
2  x.  A ) ) ) ) )

Proof of Theorem quad2
StepHypRef Expression
1 2cn 10495 . . . . . . . 8  |-  2  e.  CC
2 quad.a . . . . . . . 8  |-  ( ph  ->  A  e.  CC )
3 mulcl 9469 . . . . . . . 8  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( 2  x.  A
)  e.  CC )
41, 2, 3sylancr 663 . . . . . . 7  |-  ( ph  ->  ( 2  x.  A
)  e.  CC )
5 quad.x . . . . . . 7  |-  ( ph  ->  X  e.  CC )
64, 5mulcld 9509 . . . . . 6  |-  ( ph  ->  ( ( 2  x.  A )  x.  X
)  e.  CC )
7 quad.b . . . . . 6  |-  ( ph  ->  B  e.  CC )
86, 7addcld 9508 . . . . 5  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  +  B
)  e.  CC )
98sqcld 12109 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  e.  CC )
10 quad2.d . . . . 5  |-  ( ph  ->  D  e.  CC )
1110sqcld 12109 . . . 4  |-  ( ph  ->  ( D ^ 2 )  e.  CC )
129, 11subeq0ad 9832 . . 3  |-  ( ph  ->  ( ( ( ( ( ( 2  x.  A )  x.  X
)  +  B ) ^ 2 )  -  ( D ^ 2 ) )  =  0  <->  (
( ( ( 2  x.  A )  x.  X )  +  B
) ^ 2 )  =  ( D ^
2 ) ) )
135sqcld 12109 . . . . . . 7  |-  ( ph  ->  ( X ^ 2 )  e.  CC )
142, 13mulcld 9509 . . . . . 6  |-  ( ph  ->  ( A  x.  ( X ^ 2 ) )  e.  CC )
157, 5mulcld 9509 . . . . . . 7  |-  ( ph  ->  ( B  x.  X
)  e.  CC )
16 quad.c . . . . . . 7  |-  ( ph  ->  C  e.  CC )
1715, 16addcld 9508 . . . . . 6  |-  ( ph  ->  ( ( B  x.  X )  +  C
)  e.  CC )
1814, 17addcld 9508 . . . . 5  |-  ( ph  ->  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  e.  CC )
19 0cnd 9482 . . . . 5  |-  ( ph  ->  0  e.  CC )
20 4cn 10502 . . . . . 6  |-  4  e.  CC
21 mulcl 9469 . . . . . 6  |-  ( ( 4  e.  CC  /\  A  e.  CC )  ->  ( 4  x.  A
)  e.  CC )
2220, 2, 21sylancr 663 . . . . 5  |-  ( ph  ->  ( 4  x.  A
)  e.  CC )
2320a1i 11 . . . . . 6  |-  ( ph  ->  4  e.  CC )
24 4ne0 10521 . . . . . . 7  |-  4  =/=  0
2524a1i 11 . . . . . 6  |-  ( ph  ->  4  =/=  0 )
26 quad.z . . . . . 6  |-  ( ph  ->  A  =/=  0 )
2723, 2, 25, 26mulne0d 10091 . . . . 5  |-  ( ph  ->  ( 4  x.  A
)  =/=  0 )
2818, 19, 22, 27mulcand 10072 . . . 4  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) ) )  =  ( ( 4  x.  A )  x.  0 )  <->  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) )  =  0 ) )
296sqcld 12109 . . . . . . . 8  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X ) ^ 2 )  e.  CC )
306, 7mulcld 9509 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  x.  B
)  e.  CC )
31 mulcl 9469 . . . . . . . . 9  |-  ( ( 2  e.  CC  /\  ( ( ( 2  x.  A )  x.  X )  x.  B
)  e.  CC )  ->  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B
) )  e.  CC )
321, 30, 31sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) )  e.  CC )
332, 16mulcld 9509 . . . . . . . . 9  |-  ( ph  ->  ( A  x.  C
)  e.  CC )
34 mulcl 9469 . . . . . . . . 9  |-  ( ( 4  e.  CC  /\  ( A  x.  C
)  e.  CC )  ->  ( 4  x.  ( A  x.  C
) )  e.  CC )
3520, 33, 34sylancr 663 . . . . . . . 8  |-  ( ph  ->  ( 4  x.  ( A  x.  C )
)  e.  CC )
3629, 32, 35addassd 9511 . . . . . . 7  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) )  +  ( 4  x.  ( A  x.  C ) ) )  =  ( ( ( ( 2  x.  A )  x.  X
) ^ 2 )  +  ( ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) )  +  ( 4  x.  ( A  x.  C )
) ) ) )
377sqcld 12109 . . . . . . . 8  |-  ( ph  ->  ( B ^ 2 )  e.  CC )
3829, 32addcld 9508 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X ) ^
2 )  +  ( 2  x.  ( ( ( 2  x.  A
)  x.  X )  x.  B ) ) )  e.  CC )
3937, 38, 35pnncand 9861 . . . . . . 7  |-  ( ph  ->  ( ( ( B ^ 2 )  +  ( ( ( ( 2  x.  A )  x.  X ) ^
2 )  +  ( 2  x.  ( ( ( 2  x.  A
)  x.  X )  x.  B ) ) ) )  -  (
( B ^ 2 )  -  ( 4  x.  ( A  x.  C ) ) ) )  =  ( ( ( ( ( 2  x.  A )  x.  X ) ^ 2 )  +  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )  +  ( 4  x.  ( A  x.  C
) ) ) )
404, 5sqmuld 12123 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X ) ^ 2 )  =  ( ( ( 2  x.  A
) ^ 2 )  x.  ( X ^
2 ) ) )
41 sq2 12065 . . . . . . . . . . . . 13  |-  ( 2 ^ 2 )  =  4
4241a1i 11 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2 ^ 2 )  =  4 )
432sqvald 12108 . . . . . . . . . . . 12  |-  ( ph  ->  ( A ^ 2 )  =  ( A  x.  A ) )
4442, 43oveq12d 6210 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2 ^ 2 )  x.  ( A ^ 2 ) )  =  ( 4  x.  ( A  x.  A
) ) )
45 sqmul 12032 . . . . . . . . . . . 12  |-  ( ( 2  e.  CC  /\  A  e.  CC )  ->  ( ( 2  x.  A ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( A ^
2 ) ) )
461, 2, 45sylancr 663 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2  x.  A ) ^ 2 )  =  ( ( 2 ^ 2 )  x.  ( A ^
2 ) ) )
4723, 2, 2mulassd 9512 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 4  x.  A )  x.  A
)  =  ( 4  x.  ( A  x.  A ) ) )
4844, 46, 473eqtr4d 2502 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2  x.  A ) ^ 2 )  =  ( ( 4  x.  A )  x.  A ) )
4948oveq1d 6207 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2  x.  A ) ^
2 )  x.  ( X ^ 2 ) )  =  ( ( ( 4  x.  A )  x.  A )  x.  ( X ^ 2 ) ) )
5022, 2, 13mulassd 9512 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  A )  x.  ( X ^ 2 ) )  =  ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) ) )
5140, 49, 503eqtrrd 2497 . . . . . . . 8  |-  ( ph  ->  ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) )  =  ( ( ( 2  x.  A
)  x.  X ) ^ 2 ) )
5222, 15, 16adddid 9513 . . . . . . . . 9  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( B  x.  X
)  +  C ) )  =  ( ( ( 4  x.  A
)  x.  ( B  x.  X ) )  +  ( ( 4  x.  A )  x.  C ) ) )
53 2t2e4 10574 . . . . . . . . . . . . . . . . 17  |-  ( 2  x.  2 )  =  4
5453oveq1i 6202 . . . . . . . . . . . . . . . 16  |-  ( ( 2  x.  2 )  x.  A )  =  ( 4  x.  A
)
551a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  2  e.  CC )
5655, 55, 2mulassd 9512 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( 2  x.  2 )  x.  A
)  =  ( 2  x.  ( 2  x.  A ) ) )
5754, 56syl5eqr 2506 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 4  x.  A
)  =  ( 2  x.  ( 2  x.  A ) ) )
5857oveq1d 6207 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 4  x.  A )  x.  B
)  =  ( ( 2  x.  ( 2  x.  A ) )  x.  B ) )
5955, 4, 7mulassd 9512 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 2  x.  ( 2  x.  A
) )  x.  B
)  =  ( 2  x.  ( ( 2  x.  A )  x.  B ) ) )
6058, 59eqtrd 2492 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 4  x.  A )  x.  B
)  =  ( 2  x.  ( ( 2  x.  A )  x.  B ) ) )
6160oveq1d 6207 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  B )  x.  X
)  =  ( ( 2  x.  ( ( 2  x.  A )  x.  B ) )  x.  X ) )
624, 7mulcld 9509 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 2  x.  A )  x.  B
)  e.  CC )
6355, 62, 5mulassd 9512 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 2  x.  ( ( 2  x.  A )  x.  B
) )  x.  X
)  =  ( 2  x.  ( ( ( 2  x.  A )  x.  B )  x.  X ) ) )
6461, 63eqtrd 2492 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  B )  x.  X
)  =  ( 2  x.  ( ( ( 2  x.  A )  x.  B )  x.  X ) ) )
6522, 7, 5mulassd 9512 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  B )  x.  X
)  =  ( ( 4  x.  A )  x.  ( B  x.  X ) ) )
664, 7, 5mul32d 9682 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  B )  x.  X
)  =  ( ( ( 2  x.  A
)  x.  X )  x.  B ) )
6766oveq2d 6208 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  (
( ( 2  x.  A )  x.  B
)  x.  X ) )  =  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )
6864, 65, 673eqtr3d 2500 . . . . . . . . . 10  |-  ( ph  ->  ( ( 4  x.  A )  x.  ( B  x.  X )
)  =  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )
6923, 2, 16mulassd 9512 . . . . . . . . . 10  |-  ( ph  ->  ( ( 4  x.  A )  x.  C
)  =  ( 4  x.  ( A  x.  C ) ) )
7068, 69oveq12d 6210 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  ( B  x.  X
) )  +  ( ( 4  x.  A
)  x.  C ) )  =  ( ( 2  x.  ( ( ( 2  x.  A
)  x.  X )  x.  B ) )  +  ( 4  x.  ( A  x.  C
) ) ) )
7152, 70eqtrd 2492 . . . . . . . 8  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( B  x.  X
)  +  C ) )  =  ( ( 2  x.  ( ( ( 2  x.  A
)  x.  X )  x.  B ) )  +  ( 4  x.  ( A  x.  C
) ) ) )
7251, 71oveq12d 6210 . . . . . . 7  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) )  +  ( ( 4  x.  A )  x.  ( ( B  x.  X )  +  C ) ) )  =  ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B
) )  +  ( 4  x.  ( A  x.  C ) ) ) ) )
7336, 39, 723eqtr4rd 2503 . . . . . 6  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) )  +  ( ( 4  x.  A )  x.  ( ( B  x.  X )  +  C ) ) )  =  ( ( ( B ^ 2 )  +  ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) ) )  -  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) ) )
7422, 14, 17adddid 9513 . . . . . 6  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) ) )  =  ( ( ( 4  x.  A )  x.  ( A  x.  ( X ^ 2 ) ) )  +  ( ( 4  x.  A
)  x.  ( ( B  x.  X )  +  C ) ) ) )
75 binom2 12084 . . . . . . . . 9  |-  ( ( ( ( 2  x.  A )  x.  X
)  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  =  ( ( ( ( ( 2  x.  A )  x.  X ) ^ 2 )  +  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )  +  ( B ^
2 ) ) )
766, 7, 75syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  =  ( ( ( ( ( 2  x.  A )  x.  X ) ^ 2 )  +  ( 2  x.  ( ( ( 2  x.  A )  x.  X )  x.  B ) ) )  +  ( B ^
2 ) ) )
7738, 37addcomd 9674 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) )  +  ( B ^ 2 ) )  =  ( ( B ^ 2 )  +  ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) ) ) )
7876, 77eqtrd 2492 . . . . . . 7  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  =  ( ( B ^ 2 )  +  ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) ) ) )
79 quad2.2 . . . . . . 7  |-  ( ph  ->  ( D ^ 2 )  =  ( ( B ^ 2 )  -  ( 4  x.  ( A  x.  C
) ) ) )
8078, 79oveq12d 6210 . . . . . 6  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  +  B ) ^
2 )  -  ( D ^ 2 ) )  =  ( ( ( B ^ 2 )  +  ( ( ( ( 2  x.  A
)  x.  X ) ^ 2 )  +  ( 2  x.  (
( ( 2  x.  A )  x.  X
)  x.  B ) ) ) )  -  ( ( B ^
2 )  -  (
4  x.  ( A  x.  C ) ) ) ) )
8173, 74, 803eqtr4d 2502 . . . . 5  |-  ( ph  ->  ( ( 4  x.  A )  x.  (
( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) ) )  =  ( ( ( ( ( 2  x.  A )  x.  X
)  +  B ) ^ 2 )  -  ( D ^ 2 ) ) )
8222mul01d 9671 . . . . 5  |-  ( ph  ->  ( ( 4  x.  A )  x.  0 )  =  0 )
8381, 82eqeq12d 2473 . . . 4  |-  ( ph  ->  ( ( ( 4  x.  A )  x.  ( ( A  x.  ( X ^ 2 ) )  +  ( ( B  x.  X )  +  C ) ) )  =  ( ( 4  x.  A )  x.  0 )  <->  ( (
( ( ( 2  x.  A )  x.  X )  +  B
) ^ 2 )  -  ( D ^
2 ) )  =  0 ) )
8428, 83bitr3d 255 . . 3  |-  ( ph  ->  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  =  0  <-> 
( ( ( ( ( 2  x.  A
)  x.  X )  +  B ) ^
2 )  -  ( D ^ 2 ) )  =  0 ) )
856, 7subnegd 9829 . . . . 5  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  -  -u B
)  =  ( ( ( 2  x.  A
)  x.  X )  +  B ) )
8685oveq1d 6207 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B ) ^ 2 )  =  ( ( ( ( 2  x.  A )  x.  X
)  +  B ) ^ 2 ) )
8786eqeq1d 2453 . . 3  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  -  -u B ) ^
2 )  =  ( D ^ 2 )  <-> 
( ( ( ( 2  x.  A )  x.  X )  +  B ) ^ 2 )  =  ( D ^ 2 ) ) )
8812, 84, 873bitr4d 285 . 2  |-  ( ph  ->  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  =  0  <-> 
( ( ( ( 2  x.  A )  x.  X )  -  -u B ) ^ 2 )  =  ( D ^ 2 ) ) )
897negcld 9809 . . . 4  |-  ( ph  -> 
-u B  e.  CC )
906, 89subcld 9822 . . 3  |-  ( ph  ->  ( ( ( 2  x.  A )  x.  X )  -  -u B
)  e.  CC )
91 sqeqor 12083 . . 3  |-  ( ( ( ( ( 2  x.  A )  x.  X )  -  -u B
)  e.  CC  /\  D  e.  CC )  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  -  -u B ) ^
2 )  =  ( D ^ 2 )  <-> 
( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  D  \/  ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  -u D ) ) )
9290, 10, 91syl2anc 661 . 2  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  -  -u B ) ^
2 )  =  ( D ^ 2 )  <-> 
( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  D  \/  ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  -u D ) ) )
936, 89, 10subaddd 9840 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  D  <-> 
( -u B  +  D
)  =  ( ( 2  x.  A )  x.  X ) ) )
9489, 10addcld 9508 . . . . . 6  |-  ( ph  ->  ( -u B  +  D )  e.  CC )
95 2ne0 10517 . . . . . . . 8  |-  2  =/=  0
9695a1i 11 . . . . . . 7  |-  ( ph  ->  2  =/=  0 )
9755, 2, 96, 26mulne0d 10091 . . . . . 6  |-  ( ph  ->  ( 2  x.  A
)  =/=  0 )
9894, 4, 5, 97divmuld 10232 . . . . 5  |-  ( ph  ->  ( ( ( -u B  +  D )  /  ( 2  x.  A ) )  =  X  <->  ( ( 2  x.  A )  x.  X )  =  (
-u B  +  D
) ) )
99 eqcom 2460 . . . . 5  |-  ( X  =  ( ( -u B  +  D )  /  ( 2  x.  A ) )  <->  ( ( -u B  +  D )  /  ( 2  x.  A ) )  =  X )
100 eqcom 2460 . . . . 5  |-  ( (
-u B  +  D
)  =  ( ( 2  x.  A )  x.  X )  <->  ( (
2  x.  A )  x.  X )  =  ( -u B  +  D ) )
10198, 99, 1003bitr4g 288 . . . 4  |-  ( ph  ->  ( X  =  ( ( -u B  +  D )  /  (
2  x.  A ) )  <->  ( -u B  +  D )  =  ( ( 2  x.  A
)  x.  X ) ) )
10293, 101bitr4d 256 . . 3  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  D  <-> 
X  =  ( (
-u B  +  D
)  /  ( 2  x.  A ) ) ) )
10389, 10negsubd 9828 . . . . 5  |-  ( ph  ->  ( -u B  +  -u D )  =  (
-u B  -  D
) )
104103eqeq1d 2453 . . . 4  |-  ( ph  ->  ( ( -u B  +  -u D )  =  ( ( 2  x.  A )  x.  X
)  <->  ( -u B  -  D )  =  ( ( 2  x.  A
)  x.  X ) ) )
10510negcld 9809 . . . . 5  |-  ( ph  -> 
-u D  e.  CC )
1066, 89, 105subaddd 9840 . . . 4  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  -u D 
<->  ( -u B  +  -u D )  =  ( ( 2  x.  A
)  x.  X ) ) )
10789, 10subcld 9822 . . . . . 6  |-  ( ph  ->  ( -u B  -  D )  e.  CC )
108107, 4, 5, 97divmuld 10232 . . . . 5  |-  ( ph  ->  ( ( ( -u B  -  D )  /  ( 2  x.  A ) )  =  X  <->  ( ( 2  x.  A )  x.  X )  =  (
-u B  -  D
) ) )
109 eqcom 2460 . . . . 5  |-  ( X  =  ( ( -u B  -  D )  /  ( 2  x.  A ) )  <->  ( ( -u B  -  D )  /  ( 2  x.  A ) )  =  X )
110 eqcom 2460 . . . . 5  |-  ( (
-u B  -  D
)  =  ( ( 2  x.  A )  x.  X )  <->  ( (
2  x.  A )  x.  X )  =  ( -u B  -  D ) )
111108, 109, 1103bitr4g 288 . . . 4  |-  ( ph  ->  ( X  =  ( ( -u B  -  D )  /  (
2  x.  A ) )  <->  ( -u B  -  D )  =  ( ( 2  x.  A
)  x.  X ) ) )
112104, 106, 1113bitr4d 285 . . 3  |-  ( ph  ->  ( ( ( ( 2  x.  A )  x.  X )  -  -u B )  =  -u D 
<->  X  =  ( (
-u B  -  D
)  /  ( 2  x.  A ) ) ) )
113102, 112orbi12d 709 . 2  |-  ( ph  ->  ( ( ( ( ( 2  x.  A
)  x.  X )  -  -u B )  =  D  \/  ( ( ( 2  x.  A
)  x.  X )  -  -u B )  = 
-u D )  <->  ( X  =  ( ( -u B  +  D )  /  ( 2  x.  A ) )  \/  X  =  ( (
-u B  -  D
)  /  ( 2  x.  A ) ) ) ) )
11488, 92, 1133bitrd 279 1  |-  ( ph  ->  ( ( ( A  x.  ( X ^
2 ) )  +  ( ( B  x.  X )  +  C
) )  =  0  <-> 
( X  =  ( ( -u B  +  D )  /  (
2  x.  A ) )  \/  X  =  ( ( -u B  -  D )  /  (
2  x.  A ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    = wceq 1370    e. wcel 1758    =/= wne 2644  (class class class)co 6192   CCcc 9383   0cc0 9385    + caddc 9388    x. cmul 9390    - cmin 9698   -ucneg 9699    / cdiv 10096   2c2 10474   4c4 10476   ^cexp 11968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-2nd 6680  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-n0 10683  df-z 10750  df-uz 10965  df-seq 11910  df-exp 11969
This theorem is referenced by:  quad  22353  dcubic2  22357  dquartlem1  22364
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